Deriving SPX Option Implied Volatility from the Forward

Summary

The document describes how to choose the forward input when extracting implied volatility from European SPX options. It treats the forward as the primary quantity for a given maturity, with the yield rate derived from the forward and spot index level. Because dividends arrive unevenly, directly interpolating yields between nearby dividend dates can be problematic.

The forward can be inferred from put-call parity using available call and put prices at matching strikes and maturity. Alternatively, it can be estimated from spot using known dividend information and the funding rate. Once the maturity-specific forward is determined, implied volatility is found by solving the Black-76 pricing equation. The discussion gives a conceptual workflow rather than a numerical example, and it does not specify data-quality checks, curve construction details, or treatment of market frictions. Its method relies on the European exercise style of listed SPX options and suitable market inputs.

Key ideas

• Use the maturity-specific forward as the main input when extracting SPX option implied volatility.
• Treat the yield as a quantity derived from the forward and spot, incorporating dividends and funding.
• Uneven dividend timing makes simple interpolation of yields potentially unreliable.
• Put-call parity can provide the forward when suitable European call and put prices are available.
• With a forward in hand, solve the Black-76 equation for implied volatility.

Tags

# How should I decide the yield rate when calculate the spx implied volatility?


# How should I decide the yield rate when calculate the spx implied volatility?












I am wondering how the industry decides the yield rate for a certain maturity when calculating the implied volatility for the SPX option. Is it just a simple linear interpolation from the two near dividend records? Or something else?

Thanks!

## Answer by user808182 (score 2)

https://quant.stackexchange.com/a/73485

If your ultimate goal is backing out implied volatilities from SPX options for a certain maturity, you would determine the forward for that maturity as the fundamental quantity first. The yield rate is a derived quantity from the forward and the index spot containing the dividends and the funding component. Since dividends are not evenly spread over the year, interpolation of the yield rate is always problematic in my opinion.

If you happen to have both put and call prices available for a couple of strikes for a certain maturity you can determine the forward from put-call parity (since the listed SPX options are European). Or, in case you know the dividend records and the funding rate, you can try to calculate the forward from the index spot directly.

When you know the forward you simply need to solve Black76 equation for the implied volatility.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.